An improvement on the bounds of Weil exponential sums over Galois rings with some applications

Author

Ling, San ; Özbudak, Ferruh

Author_Institution

Dept. of Math., Nat. Univ. of Singapore, Singapore

Volume

50

Issue

10

fYear

2004

Firstpage

2529

Lastpage

2543

Abstract

We present an upper bound for Weil-type exponential sums over Galois rings of characteristic p² which improves on the analog of the Weil-Carlitz-Uchiyama bound for Galois rings obtained by Kumar, Helleseth, and Calderbank (1995). A more refined bound, expressed in terms of genera of function fields, and an analog of McEliece´s (1971) theorem on the divisibility of the homogeneous weights of codewords in trace codes over Z_p², are also derived. These results lead to an improvement on the estimation of the minimum distance of certain trace codes over Z_p² and the bounds on the correlation of certain nonlinear p-ary sequences.

Keywords

Galois fields; nonlinear codes; Galois ring; McEliece´s theorem; Weil exponential sum; Weil-Carlitz-Uchiyama bound; codeword; correlation; function field; homogeneous weight; nonlinear p-ary sequence; p-ary Kerdock code; trace code; upper bound; Block codes; Concatenated codes; Convolutional codes; IEEE Press; Information theory; Linear code; Maximum likelihood decoding; Turbo codes; Viterbi algorithm; Writing; $p$-ary Kerdock codes; Galois rings; McEliece's divisibility theorem; Weil-Carlitz-Uchiyama bound; exponential sums; nonlinear $p$-ary sequences;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.2004.834743

Filename

1337130